Sample Midterm 2

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In problems 1 - 6, find any x and y intercepts, asymptotes,
horizontal, vertical, or otherwise, find where the function assumes
positive values and where it assumes negative values, the places
where the first derivative is either 0 or does not exists, where the
function is increasing and decreasing, find the places where the
second derivative is 0 or doesn't exist, and where the function is
concave up and concave down. Find the local maxima and minima and
points of inflection, and sketch the graph.

7. A person is going to
make a box by taking a square piece of cardboard, which is
12 inches on a side, like the one in the picture to the
right, cutting squares out of the corners, and folding the
edges like the other figure in the picture to the right. How
big of a square should they cut out of the corners in order
to maximize the volume of the resulting box?

8. A rectangular sheet of
metal 20 feet long and 2 feet wide is bent down the middle
to form a "V". Two triangular pieces of metal are soldered
onto the ends to form a prismatic trough as in the picutre
to the right. What does the height of the triangle have to
be to maximize the volume of the trough?

9. What are the dimensions
of a right circular cylindrical can with a given surface
area which will result in the maximum volume?